Deducing additional set theoretic properties from definition of dynkin system

elementary-set-theorymeasure-theoryprobability theory

I am trying to prove certain set theoretic properties assuming dynkin system.
Definition of Dynkin system –
Let $X$ be a non empty set and let D be a collection of subsets of $X$, Then $D$ is a dynkin system if –

  1. $X \in D$ .
  2. If $A$ and $B$ belongs to $D$ and $A \subseteq B$, then $B \setminus A$ belongs to $D$.
  3. If $A_1, A_2, \dots$ is a sequence of subsets such that $ A_n \subseteq A_{n+1} $ for all $n \ge 1$ , then $\bigcup\limits_{n =1}^{\infty}A_n \in D$.

In following properties to be proved properties I don't know how they are true or please give some Counterexample. I am only interested how to prove them if intersection of two arbitrary sets assumed is non empty as I have already proved for disjoint sets.
Properties are –

  1. Assume sets $A$ and $B$ belong to $D$ and they have non empty intersection, does $A \cap B$ always belong to $D$.
  2. Assume sets $A$ and $B$ belong to $D$ and they have non empty intersection , does $A \cup B$ always belong to $D$ .

Best Answer

Let $A,B\subset X$ such that the sets $A\cap B$, $A\cap B^{\complement}$, $A^{\complement}\cap B$ and $A^{\complement}\cap B^{\complement}$ are not empty.

Let $\mathcal D:=\{\varnothing, A,A^{\complement},B, B^{\complement},X\}$.

Then $\mathcal D$ is a Dynkin-system but this with $A\cap B\notin\mathcal D$ and $A\cup B\notin\mathcal D$.

Observe that for $D_1,D_2\in\mathcal D$ we have $D_1\subseteq D_2$ if and only if one of the following conditions is satisfied:

  • $D_1=D_2$
  • $D_1=\varnothing$
  • $D_2=X$

This makes it easy to verify that $\mathcal D$ is indeed a Dynkin-system.

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